Autonomy is one of **the** major concerns during **the** planning of a space mission, whether its objective is scientiﬁc (interplanetary exploration, observations, etc.) or commercial (service in orbit). **For** space **rendezvous**, this autonomy depends on **the** on-board capacity of controlling **the** relative movement between two **spacecraft**. In **the** context of satellite servicing (trou- bleshooting, propellant refueling, orbit correction, end-of-life deorbit, etc.), **the** feasibility of such missions is also strongly linked to **the** ability of **the** guidance **and** **control** **algorithms** to account **for** all operational constraints (**for** example, thruster saturation or restrictions on **the** relative positioning between **the** vehicles) while maximizing **the** life of **the** vehicle (minimizing propellant consumption). **The** literature shows that this problem has been intensively studied since **the** early 2000s. However, **the** proposed **algorithms** are not entirely satisfactory. Some approaches, **for** example, degrade **the** constraints in order to be able to base **the** **control** algorithm on an eﬃcient optimization problem. Other methods accounting **for** **the** whole set of constraints of **the** problem are too cumbersome to be **embedded** on real computers existing in **the** spaceships.

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Abstract
Recent space missions rely more **and** more on **the** cooperation between different **spacecraft** in order to achieve a desired objective. Among **the** **spacecraft** proximity operations, **the** orbital **rendezvous** is a classical example that has generated a large amount of studies since **the** beginning of **the** space exploration. However, **the** motivations **and** objectives **for** **the** proximity operations have considerably changed. **The** need **for** higher autonomy, better security **and** lower costs prompts **for** **the** development of new guidance **and** **control** **algorithms**. **The** presence of different types of constraints **and** physical limitations also contributes to **the** increased complexity of **the** problem. In this challenging context, this dissertation represents a contribution to **the** development of new **spacecraft** guidance **and** **control** **algorithms**.

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In **the** literature, fuel-optimal **spacecraft** flying-formation problems **and** model predictive controllers related to space applications are traditionally formulated as Linear Programs (LP) (see [10, 16, 15]) thanks to discretization procedures. In **the** works of Deaconu **and** Louembet (see [5, 4]), a different approach **for** modeling **the** space constraints of **the** **rendezvous** problems was proposed. It consists in characterizing **the** periodic relative trajectories that are enclosed by polytopes defined in **the** local frame of **the** leader satellite by polynomials non-negativity constraints **and** employing a result demonstrated by Nesterov [13] to convert these polynomial constraints into linear matrix inequalities.

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I. Introduction
**For** **spacecraft** proximity operations (**spacecraft** **rendezvous**, station keeping, collision avoidance), **the** relative dynamics are often linearized **for** both propagation or **control** purposes. More specifically, when **the** magnitude of **the** relative motion of **the** **spacecraft** is small compared to its distance to **the** Earth, one linearizes **the** equations of motion, which implies solving simpler linear di fferential equations. However, no closed form solution is available **for** these equations in most cases. Exceptionally, **for** instance, Tschauner-Hempel equations **for** linearized Keplerian relative motion [1] admit an analytical solution **for** **the** transition matrix [2]. **For** **spacecraft** station keeping on geostationary Earth orbits (GEO), disturbing effects must be handled. Some models like **the** CNES Orange model [3] describe **the** orbital perturbations as **the** effect of **the** true geopotential, **the** lunisolar attraction **and** **the** Sun radiation pressure. A transition matrix is not available in this setting, except **for** **the** case when considering only **the** oblateness of **the** Earth in some cases, **for** which some analytical methods were developed **for** **the** description of **the** relative motion [4, 5]. However, those methods are not applicable **for** geostationary **spacecraft** due to **the** numerical issues coming from **the** zero inclination of **the** reference orbit.

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Impulsive zone model predictive **control** **for** **rendezvous** hovering phases
Christophe Louembet 1 , Alejandro H. González 2 **and** Paulo R. Arantes Gilz 1
Abstract— In this manuscript, an impulsive zone MPC **for**- mulation is proposed to tackle **the** problem of **the** **spacecraft** **rendezvous** **control**. **The** **control** objective is to maintain **the** follower **spacecraft** in a given subspace with respect to a leader vehicle by stabilizing **the** set of periodic relative orbits included in a given hovering zone. **The** idea is to incorporate this hov- ering zone as a target set into **the** MPC cost function, in order to permit a single MPC formulation **and** a receding horizon implementation. **The** **control** algorithm takes advantages of a relative motion parametrization **for** which **the** set of **the** equilibrium states represent **the** set of periodic orbits to prove **the** stability of **the** hovering zone **and** to enlarge significantly **the** domain of attraction. Several simulation results show that, in addition, performances in terms of convergence **and** fuel consumption are improved in comparison with previous works.

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Future works should focus in investigating **the** robustness of **the** proposed controller from a theoretical point of view, providing, **for** example, an idea of **the** influence of **the** nonlinearities, disturbances **and** scenario parameters on **the** stability of **the** method.
An extension of this work could combine safety requirements such as collision avoidance, passive safety or visibility, with our proposed station-keeping algorithm. This is due to **the** fact that existing guidance **algorithms** [ 35 ] which handle these constraints have **the** same mathematical formalism as **the** constrained optimization presented in this work. This can be done by considering time-varying path constraints, but this would imply revisiting **the** stability analysis while accounting **for** such time-varying constraints.

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Keywords: satellite **control**, time-varying systems, Lyapunov stability
1. INTRODUCTION
**The** last couple of decades have seen an astonishing series of achievements in aerospace science **and** technology, such as **the** increased deployment of reusable launch vehicles **and** nano-satellites, that have marked **the** beginning of an exciting new space era. These developments have led to **the** definition of new mission scenarios that necessitate more efficient hardware components **and** robust **algorithms** which however should not increase **the** overall system com- plexity **and** cost. Such requirements influence directly one of **the** most critical components **for** **the** precise operation of a **spacecraft**. This is **the** attitude **control** system (ACS) that ensures **the** active attitude stabilization **and** distur- bance rejection. **The** first immediate task of **the** ACS after launch is to detumble **the** **spacecraft**, i.e. drive all angular velocities to zero.

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Abdelkrim Achaibou ‡
LAAS du CNRS, Toulouse, FRANCE
**The** delegation to **the** flight crew of some tasks currently performed by air traffic controllers provides new perspectives to potentially increase air traffic **control** efficiency. More specifically, **the** task of establishing properly spaced landing sequences is very demanding in heavy traffic conditions **for** **the** air traffic controllers in charge of **the** terminal maneuvering area. Automatic merging **and** station keeping operations could relieve air traffic controller of providing time consuming radar vectoring instructions. **The** objective of this communication is to provide technical insight into **the** airborne devices **and** **algorithms** which may be used onboard aircraft to automatically achieve a specified distance or delay with respect to another aircraft at a specified meter fix. A nonlinear **control** law based on sliding mode **control** is proposed to **control** **the** lateral motion of **the** trailing aircraft. **The** design is followed by two illustrative examples which show **the** effectiveness of **the** proposed approach.

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commands **the** local actuators (u L **and** u F ):
u L = K L (s) · y L , u F = K F (s) · y F
Decentralized **control** is a vast ﬁeld of ongoing research which has also reached **the** area of formation ﬂying **control**, cf. [1]. **The** problem of ﬁnding an optimal decentralized controller in **the** sense of a H 2 criterion usually involves solving a linear matrix inequality (LMI) problem subject to bilinear matrix inequality (BMI) constraints, which is a non-convex problem **and** thus generally leads to local minima, cf. [8]. Although **algorithms** **for** **the** solution of this type of problem exist, **the** handling of BMI problems remains intricate. Under some conditions concerning **the** interconnection structure of **the** system **and** **the** controller, **the** problem may become convex, cf. [7]. Another possibility is to directly optimize **the** controller parameters, as described in Ref. [4], an approach which does not, alas, remedy **the** problem of local minima.

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A.3 Software
Software Structure
This section briefly outlines **the** sequence of **the** flight software implemented on **the** onboard computers **for** **the** two UAVs. When **the** computer is turned on, a RAM-disk is allotted **for** **the** storage space of **the** flight data. **The** flight software starts next **and** performs a series of initializations. One of **the** important task in **the** initialization steps is **the** declaration of a software timer interrupt. Then **the** flight software follows **the** sequence outlined in Figure A- 11. **The** program enters **the** main loop in which **the** flight software waits **for** **the** GPS data update that occurs at 5 Hz. If **the** GPS data is updated, a few functions are executed such as flight path generation, transceiver **and** copilot inputs handling, data storage, etc. On **the** other hand, after **the** definition of **the** timer interrupt, at every 0.025 seconds (40Hz) **the** timer interrupt defined in **the** initialization step begins to occur **and** a few time-critical tasks are performed. These include **the** reading **the** analog flight sensors, **the** performing **the** estimation **and** **the** controller **algorithms**, **and** **the** sending **the** **control** inputs to **the** servo motors. **The** program is terminated by **the** copilot input which is picked in **the** main loop of **the** software, **and** **the** flight data is copied from **the** RAM-disk to **the** DiskOnChip of **the** CPU module.

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To overcome these drawbacks, an event-triggered model predictive controller is considered in this work **for** station- keeping. Event-based **control** is a **control** methodology where **the** commands are asynchronously computed, re- ducing **the** communication needs between **the** sensors, **the** on-board computer **and** **the** actuators in **the** **control** loop (see Astr¨ om (2008) **for** **the** basics). This methodology can be combined with feedback policies, see Wu et al. (2014) **and** references therein, **and** model-predictive schemes, see Pawlowski et al. (2015). In **the** context of **spacecraft** oper- ations, event based controllers are recently attracting **the** attention of **the** attitude **control** community, see Wu et al. (2018) **and** Zhang et al. (2018), whereas some initial work **for** **rendezvous** hovering phases can be found in Louembet **and** Arantes Gilz (2018).

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knowledge of how some subsystem parameters affect their own requirements **and** other subsystems specifications. Due to **the** importance of an integrated **control**/subsys- tems design methodology, many attempts have been made, mainly in structural **control** literature, since **the** publi- cation of **the** first integrated design methodologies such as those in Onoda **and** Haftka (1987), Gilbert (1988) or Messac **and** Malek (1992). These methods were based on iterative methodologies with optimization **algorithms**. Lately, other methods have been proposed such as those solved by LMI **algorithms** or with LQG methods like in Hiramoto et al. (2009) **and** Cimellaro et al. (2008) respec- tively. However, these approaches give conservative results **and** their applicability is restricted by problem dimension. Recently, a counterpart technique currently under develop- ment in ONERA Toulouse Research Center allows a more general approach (Alazard et al., 2013). Actually, this method is based on structured H ∞ synthesis **algorithms**

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scenario in practice (sec. 4).
**The** simulator here discussed is a modified version of **the** one developed by Mounir Kara-Zaitri during his PhD thesis at **the** LAAS-CNRS. **The** implemented modifications were performed in order to obtain a dedicated tool **for** simulating **and** developing **control** **algorithms** **for** **the** orbital **spacecraft** **rendezvous** in **the** case where **the** leader **spacecraft** is passive **and** **the** **control** applied on follower **spacecraft** is originally computed on **the** leader LVLH frame.

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Abstract
Formation ﬂying of multiple **spacecraft** is an enabling technology **for** many future space science missions. These future missions will, **for** example, use **the** highly coor- dinated, distributed array of vehicles **for** earth mapping interferometers **and** synthetic aperture radar. This thesis presents coordination **and** **control** **algorithms** designed **for** a ﬂeet of **spacecraft**. These **algorithms** are **embedded** in a hierarchical ﬂeet archi- tecture that includes a high-level coordinator **for** **the** ﬂeet maneuvers used to form, re-size, or re-target **the** formation conﬁguration **and** low-level controllers to generate **and** implement **the** individual **control** inputs **for** each vehicle. **The** trajectory **and** **control** problems are posed as linear programming (LP) optimizations to solve **for** **the** minimum fuel maneuvers. **The** combined result of **the** high-level coordination **and** low-level controllers is a very ﬂexible optimization framework that can be used oﬀ-line to analyze aspects of a mission design **and** in real-time as part of an on-board autonomous formation ﬂying **control** system. This thesis also investigates several crit- ical issues associated with **the** implementation of this formation ﬂying approach. In particular, modiﬁcations to **the** LP **algorithms** are presented to: include robustness to sensor noise, include actuator constraints, ensure that **the** optimization solutions are always feasible, **and** reduce **the** LP solution times. Furthermore, **the** dynamics **for** **the** **control** problem are analyzed in terms of two key issues: 1) what dynamics model should be used to specify **the** desired state to maintain a passive aperture; **and** 2) what dynamics model should be used in **the** LP to represent **the** motion about this state. Several linearized models of **the** relative dynamics are considered in this analysis, including Hill’s equations **for** circular orbits, modiﬁed linear dynamics that partially account **for** **the** J 2 eﬀects, **and** Lawden’s equations **for** eccentric orbits. **The** complete formation ﬂying **control** approach is successfully demonstrated using a nonlinear sim- ulation environment that includes realistic measurement noises, disturbances, **and** actuator nonlinearities.

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Decomposition has been recognized as an important leverage to manage architectural complexity of **the** systems to be diagnosed. Most of **the** approaches, however, focus on hierarchical decomposition [4], while decentralization has been explored less frequently. **The** majority of decentralized diagnosis methods concern discrete event systems [5, 6, 7]. In [6], **the** purpose of **the** method is to provide efficient online diagnosis to detect **and** isolate faults in large discrete event systems. [5] uses a decentralized approach to deal with **the** size of **the** model **and** to get a tractable representation of diagnosis. Along **the** same idea, [7] proposes a hierarchical framework that exploits different local decisions.

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V. TESTS ON KNOWN MATERIALS
**The** 3 Omega method, as implemented by AST-DE, was tested on several materials with known properties **for** verification. First, a sample of Rohacell which was tested by **the** Physikalische Technische Bundesanstalt in Germany, with measured conductivity of 32 mW/m∙K ± 10% **and** diffusivity of 0.26 mm 2 /s ± 15%. **The** results of measurements performed at AST are shown in Fig. 9 resulting in measured conductivity of 29.9 mW/m∙K ± 4% **and** diffusivity of 0.22 mm 2 /s ± 17%. Increasing **the** heating power applied to **the** sample in **the** measurement process by a factor of 3 resulted in measurement values consistent within a few percent. **The** high probability of consistency with a constant (>0.05) indicates an over-estimate of **the** uncertainty in **the** measurements, such that true uncertainty must be lower. In Fig. 9 green lines show **the** nominal values **for** **the** sample.

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With **the** requirement of a reliable **and** safe-by-construction controller, NMPC prob- lems are solved by **the** help of **validated** simulation methods. They are mainly based on Taylor series [13, 14] or on Runge-Kutta methods [15, 16]. **The** latter is efficient in short simulation with interval initial values **and** parameters. Moreover, it is also embed- ded into **the** constraint satisfaction problems framework [17] offering new capabilities that are **the** requirements **for** **the** synthesis of robust NMPC methods.